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# Understanding the time value of money (annuity)

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Introductory lesson on calculating time value of money and annuities for non-finance majors.

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### Understanding the time value of money (annuity)

1. 1. Chapter 3: TimeValue of Money (part2)
2. 2. A QUICK DOUBLE CHECK Calculator set to 4 decimal places Calculator set to END (2nd PMT/BGN key) Calculator is set to 1 payment/yr (P/Y)
3. 3. A quick review: a single deposit FV = PV (1 + i)n  What your money will grow to be PV = FV [1/(1 + i)n ]  What your future money is worth today Inflation adjusted interest rate:  (1+i)/(1+r) -1  Substituting i* for i when controlling for inflation
4. 4.  What will John’s \$100,000 grow to be in 15 years if he leaves it in an account earning an 8% rate of return. PV = -100,000 I/Y = 8 N = 15 CPT FV = 317,216.91
5. 5. Annuities: multiple payments Definition -- a series of equal dollar payments coming at the end of a certain time period for a specified number of time periods (n). Examples – mortgages, life insurance benefits, lottery payments, retirement payments.
6. 6. Compound Annuities Definition -- depositing an equal sum of money at the end of each time period for a certain number of periods and allowing the money to grow Example – having \$50 taken out of each paycheck and put in a Christmas account earning 9% Annual Percentage Rate.
7. 7. Future Value of an Annuity(FVA) Equation This equation is used to determine the future value of a stream of deposits/ payments (PMT) invested at a specific interest rate (i), for a specific number of periods (n) For example: the value of your 401(k) contributions.
8. 8. SOLVING FOR FUTURE VALUEOF AN ANNUITY (MULTIPLE) The future value is the unknown CPT FV
9. 9. Calculating the Future Value(FVA) of an Annuity:Assuming a \$2000 annual contributionwith a 9% rate of return, how much willan IRA be worth in 30 years?FVA = PMT {[(1.09)30 – 1]/.09}FVA = \$2000 {[13.27 - 1]/.09}FVA = \$2000 {[12.27]/.09}FVA = \$2000{136.33}FVA = \$272,610
10. 10. Financial Calculator PMT = -2000 I/Y = 9 N = 30 CPT FV = 272,615
11. 11. Solving for Future value: Each month, Anna N. deposits her paycheck (\$5,000) in an account offering a monthly interest rate of 6%. How much will Anna have in her account at the end of 1 year?
12. 12. Financial Calculator PMT = -5000 I/Y = 6 N = 12 CPT FV = \$84,349.70 at the end of one year
13. 13. Practice Problems If Jenny deposits \$1,200 each year into a savings account earning an Annual Rate of return of 2% for 15 years, how much will she have at the end of the 15 years? How much will she have if she deposits \$1,200 each month? How much will she have if she earns interest monthly?
14. 14. Yearly PMT = -\$1,200 I/Y = 2 N = 15 CPT FV= \$20,752.10
15. 15. Extreme Caution! Make double sure your time frames are consistent……..  Ifthe payment is a monthly payment; then the compounding rate of return has to be a monthly rate of return.  Example: A 15% ANNUAL rate of return is equal to a monthly rate of return of 1.25%  15/12 = 1.25
16. 16. Monthly PMT = \$-1,200 I/Y = .1667 [2/12] N = 180 [15*12] CPT FV = \$251,655.66
17. 17. Present value (moves backward) & Future value (moves forward) In real life: Winning the lottery (present value) or saving for retirement (future value)
18. 18. Present Value of an Annuity(PVA) Equation This equation is used to determine the present value of a future stream of payments, such as your pension fund or insurance benefits.
19. 19. SOLVING FOR PRESENT VALUEOF AN ANNUITY (MULTIPLE) The Present Value is the unknown CPT PV
20. 20. Present Value of an Annuity: Anexample: AlimonyWhat is the present value of 25 annualpayments of \$50,000 offered to a soon-to-beex-wife, assuming a 5% annual discount rate?(PVA is the only unknown)PVA = PMT {[1 – (1/(1.05)25)]/.05}PVA = \$50,000 {[1 – (1/3.38)]/.05}PVA = \$50,000 {[1 – (.295)]/.05}PVA = \$50,000 {[.705]/.05}PVA = \$50,000 {14.10}PVA = \$704,697 lump sum if she takes thepay off today!
21. 21. Financial Calculators PMT = -50,000 N = 25 I/Y = 5 CPT PV = \$704,697.22
22. 22. Future Value Annuity of thatdivorce settlement 25 annual payments of \$50,000 invested @ 5% results in \$2,386,354.94 A difference of:  \$1,681,354.94
23. 23. Amortized Loans Definition -- loans that are repaid in equal periodic installments With an amortized loan the interest payment declines as your outstanding principal declines; therefore, with each payment you will be paying an increasing amount towards the principal of the loan. Examples -- car loans or home mortgages
24. 24. Solving for the PMT No more hypothetical “what ifs” You can really use this stuff!
25. 25. SOLVING FOR PAYMENT The Payment is the unknown CPT PMT
26. 26. Buying a Car With 4 Easy AnnualInstallments What are the annual payments to repay \$6,000 at 15% APR interest? (the payment is the unknown) PVA = PMT{[1 – (1/(1.15)4)]/.15} \$6,000 = PMT {[1 – (.572)]/.15} \$6,000 = PMT {[.4282/.15]} \$6,000 = PMT{2.854} \$6,000/2.854 = PMT \$2,102.31 = Annual PMT
27. 27. Financial Calculator PV = 6,000 I/Y = 15N=4 CPT PMT = -2,101.59
28. 28. Buying the same car with monthlypaymentsPVA = PMT{[1 – (1/(1.0125)48)]/.0125}\$6,000 = PMT {[1 – (.55087)]/.0125}\$6,000 = PMT {[.44913/.0125]}\$6,000 = PMT{35.93}\$6,000/{35.93} = PMT\$166.99 = monthly PMThttp://www.bankrate.com
29. 29. Extreme Caution! Make double sure your time frames are consistent……..  Ifthe payment is a monthly payment; then the compounding rate of return has to be a monthly rate of return.  Example: A 15% ANNUAL rate of return is equal to a monthly rate of return of 1.25%  15/12 = 1.25
30. 30. Buying the same car with monthlypayments: Financial Calculator PV = 6,000 I/Y = 1.25 [15/12] N = 48 [4*12] CPT PMT = \$-166.98
31. 31. Student loan payments Guestimate your total school loans…..(PVA) How many years to pay them off? (covert to monthly payments) At what interest rate? R u consolidating?
32. 32. Review:  Future value – the value, in the future, of a current investment  Formula?  Rule of 72 – estimates how long your investment will take to double at a given rate of return  Present value – today’s value of an investment received in the future  Formula?
33. 33. Review (cont’d)  Annuity – a periodic series of equal payments for a specific length of time  Future value of an annuity – the value, in the future, of a current stream of investments  Formula?  Present value of an annuity – today’s value of a stream of investments received in the future  Formula?
34. 34. Review (cont’d) Amortized loans – loans paid in equal periodic installments for a specific length of time